A shop is open 9 AM - 7 PM. The function r(t), graphed above, gives the rate at which customers arrive (in people/hour) at time t, where t measures time in hours since 9 AM. Suppose that the salespeople can serve customers at a rate of 85 people per hour. Answer the following questions:
Graph starts at (0,0). Reach Maximum at (6,110) and ends at (11,0)
A. At what time will people begin having to wait in line before getting served (because the volume of people arriving has become too great)?
equation editorEquation Editor hours after opening.
B. When is the line longest? Answer:
equation editorEquation Editor hours after opening.
C. How many people are in the line at the time when it's the longest?
Answer:
D. When does the line vanish?
Answer: hours after opening.
A. At what time will people begin having to wait in line before getting served (because the volume of people arriving has become too great)?
Well, my friend, the line starts to form when the rate at which people are arriving exceeds the rate at which they are being served. Using the given information, we can see that the salespeople can serve customers at a rate of 85 people per hour. So, when the rate at which people are arriving becomes greater than 85 people per hour, the waiting game begins. Looking at the graph, we can see that the volume of people (according to the function r(t)) is highest at point (6, 110). Therefore, people will start to wait in line at 6 hours after the shop opens, because after that point, more than 85 people per hour will be arriving.
B. When is the line longest?
Ah, the infamous long line! The line is at its longest when the volume of people waiting reaches its peak. Let's take a closer look at our graph. We can see that the peak volume of people occurs at point (6, 110). So, my friend, the line will be longest at 6 hours after the shop opens. That's when the wait becomes real!
C. How many people are in the line at the time when it's the longest?
Ah, the most important question of them all! At the peak of the line, at 6 hours after the shop opens, we can see that the volume of people is 110 (according to the graph). So, my friend, at the time when the line is longest, there will be 110 impatient souls waiting to be served. Hopefully, they have some good jokes to keep themselves entertained!
D. When does the line vanish?
Ah, the sweet relief of a vanished line! The line will disappear when the volume of people becomes zero. Looking at our graph, we can see that this happens at point (11, 0). Therefore, my friend, the line will vanish at 11 hours after the shop opens. That's when everyone can finally breathe a sigh of relief and get their purchases without having to wait. Hooray for short lines!
A. To find the time when people begin having to wait in line before getting served, we need to find the point where the rate of people arriving (r(t)) exceeds the rate at which salespeople can serve customers (85 people per hour).
From the graph, we can see that the rate of people arriving (r(t)) reaches its maximum at (6, 110). Since the graph starts at (0, 0), we know that the rate at 0 hours is 0. Therefore, the rate of people arriving increases from 0 to 110 in 6 hours.
So, people will begin having to wait in line before getting served when the rate of people arriving exceeds the rate at which salespeople can serve customers. This occurs 6 hours after opening, which is:
6 + 9 = 15 hours after opening.
Therefore, people will begin having to wait in line before getting served at 3 PM.
B. The line will be longest when the rate of people arriving is still high, but the rate of people being served by salespeople decreases. This occurs at the end of the graph, where the rate of people arriving is close to 0 and the rate of people being served decreases.
From the graph, we can see that the line is longest at the point (11, 0). Since the graph ends at (11, 0), we know that the rate of people arriving is 0 at 11 hours after opening.
So, the line is longest 11 hours after opening, which is:
11 + 9 = 20 hours after opening.
Therefore, the line is longest at 5 PM.
C. The number of people in the line at the time when it's the longest can be determined by subtracting the number of people being served from the number of people arriving.
At 5 PM (20 hours after opening), the rate of people being served is 85 people per hour. Therefore, the number of people being served at this time is:
85 people/hour x 20 hours = 1700 people.
Since the rate of people arriving is 0 at this time, the number of people in the line is equal to the number of people being served, which is:
1700 people.
Therefore, there are 1700 people in the line at the time when it's the longest.
D. The line will vanish when the rate of people arriving is equal to or less than the rate at which salespeople can serve customers. From the graph, we can see that the rate of people arriving becomes 0 at 11 hours after opening.
Therefore, the line will vanish 11 hours after opening, which is:
11 + 9 = 20 hours after opening.
Therefore, the line will vanish at 5 PM.
To answer these questions, we need to analyze the graph and the information provided.
A. At what time will people begin having to wait in line before getting served?
To determine the time when people begin having to wait in line, we need to find the point on the graph where the rate of customer arrivals exceeds the serving rate of 85 people per hour.
From the graph, we can see that the customer arrival rate reaches its maximum at (6, 110) and starts decreasing afterwards. Therefore, after 6 hours since the shop opened at 9 AM, the volume of people arriving becomes too great and people will have to wait in line before getting served.
Since 6 hours have passed since 9 AM, the answer is 9 AM + 6 hours = 3 PM. So, people will begin having to wait in line at 3 PM.
B. When is the line longest?
To determine when the line is longest, we need to find the point on the graph where the number of people arriving per hour exceeds the serving rate of 85 people per hour for the longest duration.
From the graph, we can see that the customer arrival rate exceeds 85 people per hour for a continuous duration from approximately 0 to 11 hours since the shop opened at 9 AM. Therefore, the line is longest between the opening time (9 AM) and the closing time (7 PM).
In terms of hours after opening, the line is longest for the entire duration of shop operation, which is 7 PM - 9 AM = 10 hours. So, the line is longest for 10 hours after opening.
C. How many people are in the line at the time when it's the longest?
To determine the number of people in the line at the time when it's the longest, we need to calculate the area under the graph between the opening time (9 AM) and the closing time (7 PM).
Since the graph starts at (0,0), reaches its maximum at (6,110), and ends at (11,0), we can divide the graph into two triangles and one rectangle.
Triangle 1: Base = 6 hours, Height = 110 people/hour
Triangle 2: Base = 1 hour, Height = 110 people/hour
Rectangle: Width = 4 hours (from 7 PM - 3 PM), Height = 110 people/hour
The total area under the graph is:
Area = (1/2) * 6 * 110 + (1/2) * 1 * 110 + 4 * 110 = 330 + 55 + 440 = 825
Therefore, at the time when the line is longest, there are 825 people in the line.
D. When does the line vanish?
To determine when the line vanishes, we need to find the point on the graph where the customer arrival rate becomes zero.
From the graph, we can see that the customer arrival rate becomes zero at (11,0). Therefore, the line vanishes after 11 hours since the shop opened at 9 AM.
Since 11 hours have passed since 9 AM, the answer is 9 AM + 11 hours = 8 PM. So, the line will vanish at 8 PM.
A. looks like you need to find where r(t) > 85
B. probably where r(t) is a max
C. the area under the curve from A to t : r(t)-85t
they are backing up during this interval.
D. See C, where the area under r(t)-85t = 0
Actually, line length is hard to say. Just because they can serve 85 per hour does not mean that they can serve them all at once, right? I suspect there is a bit more to this than a simple graph can capture.